 You can follow along with this presentation by going to nanohub.org and downloading the corresponding slides. Enjoy the show. Okay, so I'm going to be giving my last lecture now, and then I'm going to be looking forward to hearing Professor Alam talk about solar cells this afternoon. So this is a lecture, it's really not intended to be about graphene. I know that that's a very hot research topic right now, a lot of people are working on it. My intent here is really not to talk about graphene and why it's of interest and what it might be useful for, but it's really just to use it as an example to show when a new research topic emerges, you have to figure out how to understand what's going on here. You might read about graphene and you hear a lot of talk about relativistic Dirac particles and you might think that this is a very complicated material and it's difficult to understand. But what I want to show you is that most of what you need to know about electronic transport for building devices in graphene can be understood very simply with the concepts that we've introduced. It's just a different material with a different dispersion, you work everything out, we get all of the answers that we need. So let's talk about how this works. Just by way of review, these are our fundamental equations. So we're back with electrons now, not phonons. These are the expressions we've developed for the flow of electrical current and the flow of heat due to electrons. We have all these conventional expressions and we could express them as diffusive transport equations too just by having long samples. We can express them in one of two ways depending on what we want to make the independent variables gradient and quasi-firmly level and temperature or current and gradient and temperature. So we get these standard expressions that are really very, very widely applicable. So we want to apply them to a very different material now. So in order to compute conductivity in Pelletier coefficient and all of these parameters, we just need to know two things if we have a new material. We need to know what is its density of channels and what is its transmission or its mean free path. Then everything works out. And we've been discussing when we worked out the density of channels for a two-dimensional conductor, we worked out this expression a few days ago, it's got an effective mass in it. The thing to remember, first of all, is that we assumed parabolic bands when we did this and if you work it out in 1D or 3D and still assume parabolic bands, you get different answers, but there's no need to assume parabolic bands. We can work out the answer for the number of channels if we have a dispersion. So what happens then? Graphene is a really good example because it has a very simple dispersion so that we can work out things analytically, but it's very different from the parabolic band. So that's primarily why I'm doing it. And of course, a lot of people are interested in it. As you know, the Nobel Prize in Physics this year was awarded to the pioneers in graphene. A lot of people are interested not only in graphene science but in potential graphene applications. Several of you have talked to me or tell me you're working on your thesis topics involve graphene. So it's worth knowing something about this material. So the objective is just to basically to try to understand the conductance in graphene. See what people measure. So I'll remind you, graphene is just a one atom thick planar sheet of carbon with this hexagonal honeycomb lattice. You can work out its band structure. If you run this little tool in the nano hub, you can compute its dispersion and E of K in density of states and things like that. But it has a really unusual band structure. So if you do a simple tight binding model and compute E of K in the two-dimensional Brillouin zone for graphene and you get an answer that looks like this on the left. Now, if I look at that KXKY plane, that two-dimensional Brillouin zone, it turns out that there are only two distinct points at the corners of that Brillouin zone. The other ones can all be reached by a reciprocal lattice vector so they're not distinct. So the red ones and the blue ones are distinct. And I can reach the other red ones or the other blue ones just by translating by a reciprocal lattice vector so they're not distinct. So when we talk about valley degeneracy, G will be two for graphene. The states that I'm going to be interested in, the Fermi level lies at zero for a pure piece of graphene in equilibrium, charge neutral. So I'm going to be interested at the states where that top and bottom layers are intersecting because that's about E equals zero. Most of the action is going to occur near the Fermi energy. So it's going to occur near those points. There are six of those points, but only two of them are distinct. So if I look near those points, the dispersion looks very simple. The EK looks like this. It's a cone. So there's a plane, they have a Kx and a Ky coming out of the page. The E of K looks like that cone. The Fermi level ends up being right at the intersection of those two cones. I can describe that very simply. It's just plus or minus h bar of velocity. We'll call the Fermi velocity times the magnitude of K. Magnitude of K is just square root of Kx squared plus Ky squared. So the velocity, I just take the slope of the EK. The velocity of an electron is going to be Vf. And what's interesting here is that an unusual in graphene is that the slope is constant everywhere. Not if I get way, you saw in the earlier picture, if I get very far away from this point, the velocity starts to change. But under practical conditions, I'm not going to get very far away from this point. So this linear dispersion is very good. The velocity is constant. It's about 1 times 10 to the 8 centimeters per second. That's very high. That's about 10 times higher than the average thermal velocity of electrons in silicon, so that's one of the reasons that people are very interested in it. It has a valid degeneracy of 2, which I'll have to put in when I do these things. And this point, if I locate the Fermi level at the intersection of those two cones, then everything is neutral. That's called the neutral point. It's also sometimes called the Dirac point. I'll think about the states above that point as the conduction band and the states below that point as the valence band, but there's no band gap. That's also what makes it unusual. Now, so people would call this, this would be called a semi-metal. In a metal, the Fermi level is deep inside the band and there are lots of states. In a semiconductor, there's a band gap. Here, we have states very close to the Fermi level, but exactly zero if I locate them at that point. All right. Okay, so I'll mention something that I'm not going to do very much with here. If I have a normal parabolic band structure, I'll have a band gap and I'll have a conduction band and a valence band and energy will go parabolicly with k. And when I need to do things like calculate densities of states and scattering rates and things, I'll need to know the electron wave function. And we usually think of the electron as a plane wave. We would think of it as a plane wave propagating in this 2D plane. So my electron wave function would have a form like this. The square root of area is to normalize it so that when I integrate psi star psi over an area, I get 1. And but now one of the things that happens is that when the band gap gets very small that I really have to think of a multi-component wave function. The wave function has contributions from both the conduction band and the valence band. In silicon, there's light holes, heavy holes, and a split off band. So it's really a four-component wave function in which all of those components are coupled. Sometimes we have to deal with that, especially when we deal with small band gap semiconductors when the conduction band is close to the valence band and everything is coupled. Frequently in a large band gap material like silicon, there isn't a tight coupling between the conduction and valence bands and I can just use this wave function for the electrons in the conduction band. But graphene is different. The conduction and valence bands are very close together. So you would think that you always have to think about a coupled wave function that has a character due to the conduction band and a character due to the valence band. So everything is symmetrical here. That's one of the things that makes graphene so nice. Conduction band and the valence bands are mirror images of each other. The wave function, you can see that it's a two-component vector. The first one just looks like a plane wave. But the second one depends on the angle that the electron is propagating at in the x, y plane. And if you look, if the angle is zero and I'm propagating in the x direction, then those two components are both one. If the angle is pi and it's propagating in the minus x direction, then the second one has a minus one. Now, those two wave functions are orthogonal. If I take integral of psi one times psi two and integrate them, I'll get zero because they're orthogonal. And that leads to some interesting things, that sometimes people call this climb tunneling. Backscattering is suppressed because it's not possible for an electron to flip around because a scattering potential can't turn an electron exactly around because those two wave functions have no overlap. That happens in carbon nanotubes too. So that promotes higher mobilities, right? Now it's not quite as good as that and maybe people make a little bigger deal out of it than we really should for device work. Because they're only orthogonal if you exactly backscatter. But you have to take this into account when you're computing mobilities and it ends up I think if I remember right when you take this two component nature into effect, you get something like a factor of two higher mobility than you would if you ignored it. Because backscattering is suppressed completely for one particular angle but for the other ones it's reduced. Okay, now I had earlier there pointed you to a couple of lectures on Professor Dada on the nanohub. So if you'd like to see a careful discussion of band structure of graphene and how those plots are made I'd refer you to those. But that's about all we need for this lecture. So we need the number of modes, but the number of modes is the density of states times velocity. So the first thing we can talk about is the density of states. Okay, so this is going to be, this is a very simple dispersion now, but it's distinctly non parabolic. Doesn't matter, right? Because we can do a density of states. Now remind you the way that we would do that is we would look in the XY plane at a constant energy surface that would be a constant K. And we would ask how many states are there in that shaded ring? How many between NFK, DK in that shaded ring? Well the area of that shaded ring is the circumference 2 pi K times the VK. And if you remember when you impose boundary conditions on the wave function and count states, each state takes up a space 2 pi over length in the x direction and 2 pi over length in the y direction. So that's how much space each state takes up in the XY plane. So we divide that into the area, we get the total number of states. We multiply by 2 for spin and then we multiply by the valley degeneracy, which happens to be 2 for graphene also. So we get this expression for the number of states in that shaded ring. It's got a KDK. If I'd like to ask the question how many states are there between energy E and E plus DE, then I just have to do a change of variables from K to energy. And there I'll use the dispersion, E is equal to H bar VF times K. So I can do that change of variables. And if I do that change of variables, now what I have is the density of states. I'm multiplying by an A because it's the density of states per unit area, per joule, for graphene and it's EDE. So that's my answer. So if I plot the density of states above the drag point, it increases linearly with energy. If I plot the density of states below the drag point, it increases linearly with energy. Remember, for a 2D parabolic band, the density of states is constant and independent of energy. But that depends on that particular dispersion. For graphene, the density of states increases linearly with energy. All right, so density of states depends both on dimensionality and dispersion. Now how do we get the carrier density? We just integrate the density of states times the Fermi function over all the energies. Now one of the things that's nice about graphene is under most conditions, it's strongly degenerate. Once I get the Fermi level up away from that direct point, I'm deep inside the conduction band or deep inside the valence band. So the T equals 0 approximation actually works very well even at room temperature. So that simplifies everything. I don't have to deal with Fermi direct integrals very much. So I can use T equals 0, and I can ask, what's the electron density if my Fermi energy is up there? And we just do that integral. And now F is 1 between 0 and the Fermi energy, and it's 0 above that. So I only have to integrate up to EF. And the density of states goes linearly with energy. So I'm integrating EDE, and that just gives me an E squared over 2. So I get a simple expression that relates the sheet carrier density per square centimeter to the Fermi energy. It just goes as a square of the Fermi energy. And that works well even at room temperature. Okay, all right. So how do I get the number of modes, and then I can get the conductance? So remember, the number of modes, we have this general expression that works for any dispersion. It's the density of states. And we now know that times the average velocity in the direction of transport times H over 4, and then it's proportional to the width of the sample. So for graphene, we have the density of states. The velocity is independent of energy in graphene. That's because of its unusual dispersion, the linear dispersion. But I still have to average over angle, and we've done that a few times. So when I do that average over angle, just to get the x directed velocity, it's 2 over pi times the magnitude of the velocity. And since the magnitude of the velocity is independent of energy, when I multiply out and find what the number of channels is at a particular energy, it is also going to increase linearly with energy. So I have these two expressions for density of states. And for a number of channels, they both increase linearly with energy. That's a little different than when I did it for parabolic bands, I had different energy dependencies of the density of states and the number of modes because the velocity was energy dependent. In this case, I don't. Things are even simpler. The graphene is just like silicon, but a lot simpler. You can use t equals 0, calculations, things are easy. Okay, so we want conductance. There's our general expression. The second one is for near equilibrium. We know the Fermi function, transmission is going to be the same. We have to talk a little bit about what the mean free path might be for graphene. We know the number of channels, so it's easy to compute this. When I do the computation, I'm going to assume t equals 0, even at room temperature. So that derivative minus the FDE is just a delta function at the Fermi energy. So I can get simple expressions that work pretty well. So the expression is just, technically, I made the 0k assumption, but in practice it works pretty good at room temperature as well. 2q squared over h times the transmission at the Fermi energy times the number of channels at the Fermi energy. So if I'm able to, say, vary the Fermi energy up or down, I could modulate the conductance. So when I push the Fermi level up in the conduction band, I'll get a conductance that just increases linearly with the Fermi energy, assuming that the transmission doesn't change with energy too much. And if I go down, I'll also have states there. So I'll get a conductance that increases linearly with Fermi energy also. Now, in practice, the way you control the Fermi energy is by controlling the electron density. So usually people have a gate and they can control n and sub s, not the Fermi energy. So the plot will usually be g versus n sub s. So what should that look like? Well, if I plot it versus n sub s, we already determined that n sub s goes as the square of the Fermi energy. So that means that the number of channels, which is proportional to Fermi energy, goes as a square root of n sub s. So if I were to plot the expected conductance versus carrier density, which is what I can do in the lab, I would get a characteristic that looks nonlinear, looks like that. Assuming that the transmission is about constant, if it's got some strong energy dependence, it'll change this. So these are our expected results. If I go to a finite temperature, I'm going to have some spread of the Fermi function. Once I get away from that Dirac point, that spread really isn't going to matter because it's relatively small and things are not changing abruptly there. It could matter a lot right near the Dirac point. But as I'll mention a little bit later, we need to stay away from the Dirac point because things get very messy and complicated there. Near the Dirac point, if I put the Fermi level at the Dirac point, my t equals 0 assumption will say there are no states there, so there is no conductance there. Well, there's a little bit of thermal spread, so I'll get some states above the Dirac point contributing in some states below. But in practice, something else happens that is even more important. So this is not too much of a detail to worry about. All right, so at finite temperature in an ideal system, the red line is what I would expect to get. Okay, so these are the key equations. The first one describes the conductance. Whether the Fermi level is above or below the conduction band, we can, as long as we're away from the Dirac point, we can describe the conductance with that expression. Technically, for t greater than 0, I would have to work out a Fermi-Dirac integral and I have contributions from both bands. I'm going to have to talk about what the mean free path might be. People usually like to express this in terms of a sheet conductance. So you just express the conductance as sheet conductance times W over L. And what's usually reported in experiments is the sheet conductance. So the sheet conductance, if we work it out like we've done before, would be given by this expression. Where lambda apparent, remember, is the actual mean free path if I have a diffusive sample. But if I have a sample that may be between the ballistic and diffusive regime, it's the smaller of the length of the channel and the physical mean free path. This is actually a little bit important in graphene, because it has very high mean, long mean free paths. And you can see these ballistic effects. Okay, so let me talk a little bit about scattering. So if we want to understand this in more detail, we have to understand something about the mean free path. Remember, it's a mean free path for back scattering. So we have to understand something about scattering and graphene. So as I mentioned earlier, but for many scattering mechanisms, the way the scattering works is an electron comes in at some energy E. It gets scattered to some final energy. If the scattering is elastic, an acoustic phonons, the ones that scatter electrons most strongly are near the center of the Brillouin zone and they don't carry a lot of energy. So it's almost elastic. So we would expect the probability that you'll scatter per unit time one over tau to be proportional to the number of states that you can scatter to. So it should be proportional to the density of states. The density of states is proportional to energy. That means the scattering time for a mechanism like this, like acoustic phonon scattering, would be proportional to one over energy. So as you get higher and higher away from the direct point and you have more and more states, the time, the scattering time decreases because there's more and more states to scatter to. So our energy mean free path would be proportional to one over, our energy dependent mean free path would be proportional to one over energy. Now it's interesting to ask what would this do, what kind of conductance would we measure if this were the dominant scattering mechanism? Because again, it's straightforward to calculate, but it's a little surprising when you see the answer. But it just comes from this unusual dispersion of silicon. So this is our expression for, let's say, diffusive transport. That's our expression for the sheet conductance. 2q squared over h times mean free path times a number of channels at the Fermi energy. The mean free path for this type of scattering is proportional to one over Fermi energy. The number of channels is proportional to Fermi energy. So the conductance is constant. So this is not usually what we expect. Usually you expect if you have more electrons in the conduction band, you'll have lower conductance. What we have for graphene is it's independent of the number of electrons. If acoustic phonon scattering is dominated, you put twice as many electrons in, you get exactly the same conductance. That happens because you have twice as much scattering. You put in twice as many, you get twice as much scattering. Your mean free path is halved and you get the same conductance. So that's a very unusual thing that was pointed out several years ago. So you could ask yourself, and I'll refer you to some notes later, you can work out these scattering times for acoustic phonon scattering. And you can put in actual numbers if you know things like deformation potentials, and you could compute what is the conductance of graphene if you just have acoustic phonon scattering. And it turns out that putting in numbers at room temperature, it's about 30 ohms per square. So I think there was a question yesterday about transparent conductors for solar cells or something like that. And people who design solar cells, they like a sheet resistance on the order of 10 ohms per square so that they get good performance. So you can see that this is sort of an intrinsic sheet conductance of graphene because you're always going to have some acoustic phonon scattering at your temperature, the best you can do is 30 ohms per square. So you'd have to do multiple layers of graphene to get its sheet resistance down to a place that you could use it for an application like that. Now in many of these samples, the resistivity is much higher because there are things like charged impurities that can scatter electrons. So let's say I'm looking down on the top. I put a sheet of graphene on a silicon dioxide layer, which is sitting on a wafer. I've made a resistor here, but there are stray charges around. They may be charges in the oxide. They may be charges in the ambient or due to the processing that has been left over. And those stray charges can scatter electrons. So that's the charged impurity scattering that we've talked about a couple of times. So that's going to fluctuate the direct point up and down. It's going to introduce this scattering potential. And again, whenever I have a scattering potential like this, the high energy carriers aren't going to see those fluctuations in potential. They're not going to be so important. So the high energy carriers will scatter less. So this would give me a mean free path that would increase as the energy increases. And actually, if you work out this scattering rate and you do it carefully, you'll get some complicated expressions. But you'll find that in the end that the mean free path is approximately proportional to the energy. The higher the energy, the longer the mean free path. When you're dominated by this type of scattering. So if I go back and ask what would that do to the measured sheet conductance, I start with 2q squared over h times a mean free path times a number of channels that are occupied at the Fermi energy. I now have a mean free path that's proportional to the Fermi energy. That means my conductance is proportional to the Fermi energy squared. Remember that the carrier density was proportional to the Fermi energy squared. For this particular scattering mechanism, the conductance is proportional to the carrier density, which is what you would expect. So you'd expect to see a constant mobility if you do a calculation like this. Okay, so let's talk a little bit about how you do these experiments and look at some data. So in practice, people do near equilibrium transport measurements of graphene to try to understand the properties of graphene and how good their graphene is. The way they'll do it is that they will use a gate to move the Fermi energy up and down. And typically in the experiment, you have a layer of graphene on a silicon dioxide layer, you attach two contacts to it, or maybe four, so you can do four probe measurements, you use the electrode on the bottom as a gate to modulate the position of the Fermi level. And what you really control there is the electron density, which is related to the Fermi energy. So this is what it looks like from a side view. There's a layer of graphene on top of a silicon dioxide layer. It's usually something like 300 nanometers thick because if you put one monolayer of graphene on a 300 nanometer thick layer of SiO2 and illuminate it with typical illumination, the interference patterns will be set up in a way that you can see one monolayer of graphene and you can tell you've got it there. So 90 nanometers, I think you can also see it. So you tend to see the layers in one of those two dimensions. Okay, all right, so the idea is we're going to measure the voltage. This would be a two probe measurement that might suffer from contacts, but you could do four probe measurements as well. We use the bottom contact as a gate and we gate it from the bottom. And because of that, I'm not actually showing. We don't actually move the Fermi level. The Fermi level stays constant. It's fixed by the contacts. But we actually move the Dirac point up and down. So if we apply a positive voltage, we would lower the Dirac point and that would put the Fermi level in the conduction band. If we would apply a negative gate voltage, that would pull the Dirac point up and the Fermi level would be in the valence band. So we can sweep the valence band through the Dirac point and measure the conductance. So if you did the measurement, you would get something like this. The minimum, you could identify the minimum and say, well, that's where the Dirac point is at that voltage. It won't typically occur at zero voltage because there will be some work function difference between the gate and the graphene. And there will be some straight charges that are fixed amount that are there that will shift it. So people will typically then report, they'll shift the zero to the origin and they'll report it that way. So it's easy to relate the gate voltage to the sheet carrier density because charge is just capacitance times voltage. Now you have to be a little more careful than that. The graphene has capacitance two. It's really two capacitors in series and the graphene capacitance is something people sometimes call a quantum capacitance. And you can do that calculation more carefully. What you find is that whenever the SiO2 layer is bigger than a few nanometers or so, that this is a very good approximation. The only capacitance you have to worry about is the capacitance of the insulator. So this gives people a very simple way experimentally to control the density of electrons in the graphene. And then you can study how the conductance varies. So here are some measured results. So we can see linear characteristic. We're plotting it versus gate voltage. And N sub s is capacitance times gate voltage. We see a linear characteristic. So right away we think we must be dominated by charged impurity scattering in this particular sample. Now I could go through at any point and I could take my expression. I've measured the sheet conductance and that's reported here. I know what the carrier density is because my gate voltage gives that to me. And I know the insulator capacitance. So I could deduce what the mean free path is at any point. And if you do that up here at a gate voltage of 100 and you take the measured conductance, you deduce what the sheet carrier density is. Plug numbers into that expression. That sheet carrier density will correspond to a Fermi energy of 3 tenths of an EV. So what we find is that at that particular point, the mean free path is 130 nanometers. It's much less, let's see, the length of that resistor is 5,000 nanometers. So it's much less than the length. So I'm in the diffusive regime. I don't have to worry about ballistic transport. If I go down here to a lower voltage, I can pick off the sheet conductance there. I've got a lower carrier density that I can figure out. The Fermi energy is 0.2 EV above the direct point at this point. And I can solve that expression I showed earlier and find out what the mean free path is. And it's about 90 nanometers, so it's shorter. So the ratio of the mean free path is about 69%. The ratio of the energies is about 67%. So it looks like the mean free path is increasing linearly with energy. That's what you expect for charged impurity scattering. And that's likely what's going on here. Since the conductance is proportional to the carrier density in this regime, I can deduce a mobility. If I were to deduce a mobility from the slope of that line, it's about 12,000 or so. Nice high mobility. Another reason that people are interested in graphene. Now, this region near the bottom, I mentioned that we usually try to stay away from that regime. The reason is that if you look, this is I guess a scanning tunneling electron. Using some scanning probe measurements to try to determine what the electron density is versus position in graphene. And what you find is that the graphene isn't perfect. It's got charges and variations that are moving the direct point up and down. So when you get near the direct point, you see these fluctuations. You can't assume when I gate this structure with a voltage that should put the Fermi level at the direct point, it's not going to be at the direct point everywhere. So you have to average over all of these fluctuations, the electron hole puddles. And that's why the description near the direct point gets more complicated. If I get the Fermi energy way above those potential fluctuations, then they're not so important. And the theory that I discussed works just fine. Okay, now these are some experiments where they controllably put in potassium dopants. So they introduced charged scattering centers in a controllable way. And you can see the first line there on the right is the initial sample where it wasn't exposed to phosphorous at all. And you get a characteristic that's slightly linear. And as you expose the characteristics to more and more phosphorous, the characteristics get more and more linear, and the slope gets less and less. So that's what we would expect then for charged impurities scattering is that we take an initial sample that doesn't have a lot of charged impurities. And as we increase charged impurities, they just get more and more, we just get more and more scattering and we get a lower and lower slope. Notice that the neutral point also shifts because the background charge that there is shifts the electrostatics as well. So I could look at that first one and I could say that, well, I know that way back when I did it ballistically, I told you that the ballistic characteristic should go as the square root of gate voltage or carrier density. So I could look at that initial one and I could ask how close is it to the ballistic limit? Am I dealing with a diffusive sample or a ballistic sample? So again, we had this expression here which relates to measured conductance and the carrier density that we've induced with a gate voltage to the mean free path. So I could just go there and read off data and estimate the mean free path. And it's about 164 microns. And I forget in this particular sample, I think the sample length is over a micron. So we're in a diffusive regime. So that nonlinear characteristic is not coming from ballistic transport. It's coming from some other scattering mechanisms. Now this is another experiment. This is done at Columbia. This is an initial sample. You see the blue line where the graphene was sitting on the substrate. And then they could etch out the underlying substrate so that the graphene was just suspended and they could anneal it and drive off all of the stray charges. So they had a much more pure sample. And what they see is higher conductance and the initial linear characteristic is converted into a characteristic that's nonlinear. Now you could ask on this, is it near the ballistic limit or is it near the diffusive limit? Well if you look at that length scale up there, the length of the resistor that they're measuring, you can see a four probe pattern there to do four probe measurements and Hall Effect measurements. You can see that the length is on the order of a couple of microns and the mean free path that they extract is on the order of a micron. So we're probably getting close to a ballistic transport. If we just use our expression in there, we can deduce an apparent mean free path. Notice the one that we get is a little bit longer than the one that they quoted in the paper because ours is this mean free path for back scattering, this landauer mean free path that's got the pi over 2 in it. So it's a little bit longer. But you can see that we're getting close to the dimensions of the sample. So at this regime, we're starting to approach ballistic transport. So it's possible with very clean graphene to get close to the ballistic limit. Now these are some other measurements, these are temperature dependent measurements. And now this isn't conductance, it's one over conductance. It's resistivity. And what you're seeing here on the right are plots of resistivity versus temperature for two different graphene samples. And the main point I want you to notice there is just the two different regimes. There's an initial regime where the resistivity increases linearly with temperature, and then it starts to increase much more rapidly with temperature. So for low temperatures below 100K, the resistivity or resistance is proportional to T. But above that, the resistivity starts to go exponentially with temperature. Now there you start to suspect scattering by optical phonons. Because the population of optical phonons depends on their energy, and it goes as something like H bar omega over KT. So that could be optical phonons in graphene. It turns out that people really think that what's happening is that those are optical phonons in the SiO2 underneath the graphene. Because they have the energies that seem to explain when this starts to kick up. So you can understand the general characteristics very simply. The resistance is one over the conductance. So the resistance is proportional to one over the mean free path. That means the resistance is proportional to one over tower, the scattering rate. The scattering rate should be proportional to the number of phonons that are there to scatter from. The number of phonons is given by that Bose-Einstein factor. So the resistance should be proportional to that Bose-Einstein factor. Acoustic phonons have small energy. So H bar omega tends to be small compared to KT. That means that exponential, I can expand it for small argument. And that little n and capital N are the same thing. But the n naught then becomes KT over H bar omega. That's very intuitive. Thermal energy is KT. H bar omega is the energy of the acoustic phonon. I just divide the two and I get the number. So if acoustic phonons were dominating the scattering process, I would expect the resistance to be proportional to temperature. That's what we saw initially. Optical phonons have much bigger energies. They're comparable to KT. So I can't do that Taylor series expansion of the exponential. I'm just going to expect the resistance to have that exponential characteristic. Once the thermal energy gets comparable to H bar omega, I should start to increase the population of optical phonons and I get more scattering. That's why it increases. So these are the kind of, as I mentioned, a lot of what people use near equilibrium transport measurements for is when you have a new material that you're working with. You do these kinds of measurements, you vary the temperature and you try to deduce what's going on. What is the band structure? Is it what I think it is? What scattering mechanisms are controlling the performance? And there's been a lot of work over the past three or four years where people have done a lot of these near equilibrium measurements and tried to sort this all out. And the general picture is something like this. We said, if you have a ballistic device, we worked it out and we said the conductance should go as a square root of voltage, square root of carrier density. If we're dominated by acoustic phonons scattering, let's say we have a pure sample. We're at a low enough temperature that we're not exciting optical phonons. We have no charged impurities. Then we said that the conductance is independent of gate voltage. So it's just that horizontal line. And if the conductance is dominated by charged impurity scattering, it gives a linear characteristic. And in practice, we might have some contribution to charged impurities. And we might have some contribution due to acoustic phonons. So we could expect to see some kind of nonlinear characteristic, that is sort of the sum of those two. So that initial, remember when I talked to you about that experiment with phosphorus doping, the initial one before any charged impurities were intentionally put in was a little bit nonlinear. It was probably a little bit nonlinear because we were seeing a combination of some background unintentional charged impurity scattering. But it was weak enough that we were beginning to see the limitations due to acoustic phonons scattering as well. And when we intentionally introduced more scattering centers, then we were just totally dominated by the charged impurity scattering and the other one was so higher that we didn't see it. Okay, now I should mention, if you do these two probe measurements, you'll tend to get a characteristic that looks like this. It won't be symmetrical. The conduction and valence band of graphene are perfectly symmetrical. And when people first saw this, it wasn't obvious what the difference might be. What could be causing it? There are a number of things. One of the things is if you look at how an electron scatters off of a positive charge and how an electron scatters off of a negative charge, it's different. So on one branch, we have electrons scattering and on the other branch, we have holes scattering. So the sign of the interaction is changing. If you calculate the scattering rate by Fermi's golden rule, it doesn't make any difference. But if you do the scattering calculation a little more carefully, when there's an attractive potential between the two, it scatters a little more strongly than when there's a repulsive one. So that was hypothesized and that may be part of the answer. But the main answer seems to be the asymmetries are coming from the contacts. That there's some nonlinear characteristic and the IV characteristic that comes from the contacts. So you have a different contact resistance on the N branch than you do on the P branch. You can eliminate that. You can test that hypothesis by doing a four probe measurement. And when they did a four probe measurement, they found that the characteristics were symmetrical. So most of it seems to be coming from the contacts. And that's something that people worry about here now. Okay, so all right, there's something a little interesting here. I want to just mention it because it's kind of cute. This is our expression for sheet conductance, okay? And I can always equate my sheet conductance to Nq mu. And then I can derive a theoretical expression for mobility. So let's do that. We have an expression for the number of channels. We have an expression for the mean free path. We have an expression for the carrier density. I can just solve the top equation for the mobility. And if I solve the top equation for the mobility, this is the expression I'll get. Q tau over EF squared over Fermi velocity squared. That looks like Q tau over M. It looks like the effective mass of graphene is Fermi level over BF squared. Now that's actually kind of cute. That sort of looks like a relativistic expression. E is equal to MC squared. And this has a dispersion that's like light. So that may not be unexpected. Remember, people do Hall effect measurements in graphene and they worry about the cyclotron resonance frequency. For a parabolic band, the cyclotron resonance frequency is QB over M. What is the cyclotron resonance frequency in graphene? It turns out it's QB over this M. You get the right answer for that too. So it's actually a little more than fortuitous. If you look at Professor Dada's notes, probably in the ones that he put on the flash drive. He has an argument that a better way to define effective mass for transport is that it's the ratio of momentum to velocity or crystal momentum to velocity. If you use that definition and the dispersion of graphene, you'll get this answer for the effective mass of graphene. If you use that definition of effective mass and a parabolic dispersion, you'll get the normal effective mass. So people say graphene has no effective mass, but there are cases where if you're careful about thinking about it, you can assign an appropriate effective mass to it. All right, if you're looking for a challenge, you can think about working out the Hall effect in graphene. The answer is in the notes that I put on the flash drive, but it takes a little bit of effort. Also one of the interesting features of graphene is that it has an unusually high thermal conductivity. So you could try to think about why it has a high thermal conductivity. See if you could figure that out. Okay, so that's basically it. And as I said, the point was just to show you how if you encounter a new material and you start reading papers, it can frequently look very strange. But if it's a problem in electrical transport, the first thing you want to do is to go back and say, well, can I understand these ideas in terms of the concepts that I know from other materials? And it turns out you can. So graphene has some really interesting features that come from this dispersion. But in terms of understanding all of these, all we have to do is to explain the same concepts that start with a different dispersion, then everything falls out pretty naturally. There are a lot of interesting things, applications that people are exploring these days, it's a little hard to see which ones are going to end up being important, it's likely that some of them will be. But it also makes a wonderful set of homework problems because the dispersion is so simple, it's so easy once you learn things with parabolic bands to give a homework exercise, work it out for graphene and it's easy to do analytically. Okay, if you want a lot more information about this, we actually had one of the previous summer schools. We had a set of four or five lectures on graphene, so I can refer you to those. We have a set of notes where we work out a lot of these things like densities, escapes, states, and scattering times. And my student Dionysus was a co-author on this, the lead author on this, so you can find that if you're interested in it too. Of course, there's a lot of information on graphene available in many different places. Okay, so I'll stop there and see if you have any questions. Yes, skip. When you first started talking about a lot of purities, they thought, well, maybe it's going to be an SA2 or interstitial or something like that. But then when you run into a graphene and purity, I thought I could use it in a crystal graphene glass. Yeah. So it's in the crystal graphene glass that it's possible to shift the frontal lobes with these purities. And as soon as it wasn't there, as you know, we talked about changing the shape of the structure, which I've been talking about for a long time. Yeah. I'm going to confuse these purities in the crystal structure. Yeah, so the question is where are these, these imperities, we're not doping graphene with substitutional doping, right? So the imperities that I'm speaking about here, like the phosphorous, these are not in the graphene honeycomb structure. You're coating the graphene with something on top. And the charge then is affecting the Dirac point and moving things up down. So it might be analogous to modulation doping for semiconductor people. And the background imperities, when you take a sample and put it on an SA2 layer and then measure the characteristics and you see a linear characteristic, you know, there are impurities somewhere. A lot of them tend to be in the SIO2 that you did the processing on. A lot of them tend to be left over from the ambient when you did all of the processing on the materials. You can heat it up and try to drive some of them off. But it's only one monolayer thick, so it's very sensitive to whether there are any charges in the environment close to it. Yeah. When you look at the sheet, they show sheet tend to be on the order of 20 and 20, but there is no steps. There is no what? There is no quantization steps that we expect. There is no quantity. Oh. You mean there is no quantized conductance? Yeah. So all you're right that in the characteristics I showed there was no evidence of quantized conductance. And that's because these were all very wide structures for which there were many modes. And when I go to very wide structures, then the conductance is just proportional to the width of the resistor. Now, you know, when you experimentally early on when I showed you that experiment that showed these steps, that was a very narrow conductor. And I could electrically control the width of the conductor through a shot key barrier, so I could then deplete it and I could individually count the number of modes. Now, I don't think I've seen that in graphene. If you try to make a graphene ribbon, you know, a lot of research these days is devoted to how do we induce a band gap in graphene so that we can make a better transistor out of it and turn it off. And one of the ways is to make a very narrow strip. And then you could start to count modes. And the difficulties there is you have so much roughness and disorder at the edges that it's hard. The challenge is to try to figure out ways to make smooth edges that don't introduce so much scattering and randomness that it affects the transport. Any other questions? Right down here. We know graphene consists of a number of modes and I'm learning the experience when we change some of the levels of this atom by other atoms. For example, the volume of other atoms. Yeah. So where you would substitute other experiments, where you would substitute other atoms for carbon in them. Yes, it works. Yeah. Yeah. It doesn't mean that there aren't, but I'm not aware of any. These carbon-carbon bonds are very strong so it's hard to do substitutional doping the way that you do in silicon. The limits of force. Pardon me? We have other elements of force group, which have also force. Right, but you have to have enough thermal energy to break a bond and dope it or you have to grow it in that way. And maybe people have looked at that, I'm not aware of anything. There are a variety of other materials. The thing that was nice about graphene is you could take a chunk of graphite and then you could rub it across an SIL2 layer and the individual layers would peel off. And if you learn how to do that just right, you can get one monolayer. That's what the folks who got the Nobel Prize learned how to do. There are other layered materials like this where the bonding is very strong within a layer but very weak between layers. People are interested in other materials like this where you can produce single monolayers of various materials. So there may be other examples of materials that are one monolayer thick that are of interest for electronic applications. In fact, there are. But it's just amazing how quickly this Nassau-Love and Geim learned how to do this. Let's see, they got the layers, I think they use Scotch tape to put Scotch tape on graphene and pull off one monolayer and then transfer it to an SIL2 layer. Now you can get graphene layers over 12-inch silicon wafers. It's just amazing how quickly, people have done it in roll-to-roll processing now where you can have large sheets. It's amazing how quickly it has come in just a few years. Now those layers tend to have many more defects. They're polycrystalline, they have grain boundaries, it's scattered, but the transport of the individual grain is described by concepts like this. Yes. And I like that Tuskegee, we've experimented with graphene, actually making graphene product among forgathal clay. We also can challenge some impurities with silicon, some aluminum that we try to wash away with that phenol ethyl. But using the furnace, we've tried to actually graphene with cadmium sulfide. We have like quantum dots on top of this sheet. I was wondering, by using an acoustic wave through ultrasound with that, using graphene oxide, because it does have impurities. Because the graphene that we order is a graphene powder. It's just sort of a mesh, how small, 400 nanometers or something like that. But we can't look copper as the impurity. It won't be pretty on the TEM, we said that it was 97% pure. But it wasn't actually, we had our copper along with it. So this is, you're saying you purchased graphene or you obtained it from another source? So you don't? Purchase and we obtained the cheap graphene by furnace under high temperature. So we're able to produce graphene, but it's not pure graphene. We actually have pureness coming up. And I think one of the growth techniques for this large area of graphene now involves growing it on copper. The copper catalyzes it. But then it's maybe not surprising that you find some copper in the graphene. It's not surprising, but this is from the super market graphene, the super market that we purchased. It's taking that as none, so I'm just saying you have to be careful about what you do. And so what you would probably want to do is to do some transport characterization and see what kind of mobilities you're getting. See if the characteristics look linear and if you can identify this as charged impurity scattering or something else. Thank you.